Question

Define F on R3 by F(x) = (0, 0, cx3)x and let M be a compact three-dimensional manifold-with-boundary with MC {x: x3 <0}. The vector field F may be thought of as the downward pressure of a fluid of density in {x: x2 < 0}. Since a fluid exerts equal pressures in all directions, we define the buoyant force on M, due to the fluid, as −∫∂M dA. Prove the following theorem.